Perfect numbers: Old and new issues; perspectives
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PERFECT NUMBERS: OLD AND NEW ISSUES; PERSPECTIVES
1.1
Introduction
The aim of this chapter is to survey the most important and interesting notions, results, extensions, generalizations related to perfect numbers. Many old, as well as new open problems will be stated, which will motivate - we do hope - many further research. This is one of the oldest subjects of Mathematics, with a considerable history. Some basic historical facts will be presented, as this will underline too our strong opinion on the role of perfect numbers in the development of Mathematics. It is sufficient to only mention here Fermat’s ”little theorem” of considerable importance in Number theory, Algebra, and more recently in Criptography. This theorem states that for all primes p and all positive integers a, p divides a p − a. Fermat discovered this result by studying perfect numbers, and trying to elaborate a theory of these numbers. One more example is the theory of primes in special sequences, and generally the classical theory of primes. Even perfect numbers involve the so called ”Mersenne primes”, of great importance in many parts of Number theory. Currently, about 39 such primes are known (39 as of 14-XI-2001, see e.g. http://www.stormloader.com/ajy/perfect/html), giving 39 known perfect numbers, all even. Recently (at the end of 2003) the 40th perfect number has been discovered. No odd perfect numbers are known, but we shall see on the part containing this theme, the most important and up-to-date results obtained along the centuries. An extension 15
CHAPTER 1
of perfect numbers are the ”amicable numbers” having a same old history, with considerable interest for many mathematicians. Many results, more generalizations, connections, analogies will be pointed out. Here the theory is filled again by a lot of unsolved problems. Along with the extensions of the notion of divisibility, there appeared many new notions of perfect numbers. These are e.g. the unitary perfect-, nonunitary perfect, biunitary perfect-, exponential perfect-, infinitary perfect-, hyperperfect-, integer perfect-, etc., numbers. On the other hand, there appeared also the necessity of studying, by analogy with the classical case, such notions as: superperfect-, almost perfect-, quasiperfect-, pseudoperfect (or semi-perfect), multiplicatively perfect, etc., numbers. Some authors use different terminologies, so one aim is also to fix in the literature the exact terminologies and notations. Our aim is also to include results and references from papers published in certain little known journals (or unpublished results, obtained by personal communication to the authors).
1.2
Some historical facts
It is not exactly known when perfect numbers were first introduced, but it is quite possible that the Egypteans would have come across such numbers, given the way their methods of calculation worked (”unit fractions”, ”Egyptean fractions”). These numbers were studied by their mystical properties by Pythagoras, and his followers. For the Pythagorean school the ”parts” of a number
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